# Integral exponential function

Jump to: navigation, search

The special function defined for real by the equation The graph of the integral exponential function is illustrated in Fig.. Figure: i051440a

Graphs of the functions , and .

For , the function has an infinite discontinuity at , and the integral exponential function is understood in the sense of the principal value of this integral: The integral exponential function can be represented by the series (1)

and (2)

where is the Euler constant.

There is an asymptotic representation: As a function of the complex variable , the integral exponential function is a single-valued analytic function in the -plane slit along the positive real semi-axis ; here the value of is chosen such that . The behaviour of close to the slit is described by the limiting relations: The asymptotic representation in the region is: The integral exponential function is related to the integral logarithm by the formulas  and to the integral sine and the integral cosine by the formulas: The differentiation formula is: The following notations are sometimes used:  